Користувач:Галактион/Modus tollens

Подстраница "Користувач:Галактион/Modus tollens " создана для того, чтобы перенести информацию из раздела "Обговорення" статьи "Modus tollens". Галактион 14:54, 5 березня 2010 (UTC)

Дополнение к статье "Modus tollens"

Предварительные замечания

0. Сведения подразделяются на знания (факты) и предположения (гипотезы).

Если предложению предшествует верификатор "|–" либо его аналог, то это предложение каталогизируется как элемент множества знаний (факт).

Если предложению предшествует верификатор "h |–" либо его аналог, то это предложение каталогизируется как элемент множества предположений (гипотеза).

Примечание:

В Английском языке аналогом верификатора "|–" является вводное предложение "It is known that", а аналогом "h |–" является вводное предложение "It seems that".

В Русском языке аналогом верификатора "|–" является вводное предложение "Известно, что", а аналогом "h |–" является вводное слово "Кажется".

1. Правилами логического вывода называются правила получения [новых] знаний из [известных] знаний.

Каждое правило вывода можно записать в виде:

$~\vdash {\mathfrak {a}}_{0},\quad \vdash {\mathfrak {a}}_{1},\quad ...,\quad \vdash {\mathfrak {a}}_{n}\quad \Rrightarrow \quad \vdash {\mathfrak {b}}$,

где:

1) предложения $~{\mathfrak {a}}_{0},\ {\mathfrak {a}}_{1},...,{\mathfrak {a}}_{n}$ с предшествующими им верификаторами $~\vdash$ – это известные факты,

2) предложение $~{\mathfrak {b}}$ с предшествующим ему верификатором $~\vdash$ – это новый (получаемый) факт.

3) символ $~\Rrightarrow$ – это аналог Английского слова "therefore", а также Русского словосочетания "стало быть" или Русского слова "поэтому".

2. Modus tollens – это правило вывода, которое позволяет получать из двух [известных] фактов один [новый] факт.

Примечание:

Существуют и другие правила вывода, которые позволяют получать из двух [известных] фактов один [новый] факт.

Формулировки "Modus tollens"

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash \ \neg {\mathfrak {b}}\quad \Rrightarrow \quad \vdash \ \neg {\mathfrak {a}}$

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash \ \neg {\mathfrak {b}}\quad \therefore \ \quad \vdash \ \neg {\mathfrak {a}}$

$~\vdash \ {\mathfrak {a\to b}},\quad \vdash \ \neg {\mathfrak {b}}\quad \mathrm {Therefore} \quad \vdash \ \neg {\mathfrak {a}}$

Дополнительные пояснения

Формулировка

It is known that "$~{\mathfrak {a}}.$" implies "$~{\mathfrak {b}}.$". [However,] it is known that it is false that $~{\mathfrak {b}}.$ $~\Rrightarrow$ It is known that it is false that $~{\mathfrak {a}}.$

Examples

1. It is known that [the sentence] "I don't eat varenyky." implies [the sentence] "Varenyky eats me.". However, it is known that it is false that varenyky eats me. $~\Rrightarrow$ It is known that it is false that I don't eat varenyky.

Примечание

Предложение "It is known that "I don't eat varenyky." implies "Varenyky eats me."." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложение "It is known that it is false that varenyky eats me." и предложение "It is known that it is false that I don't eat varenyky." - это предложения Английского предметного языка с верификаторами "It is known that...".

Предложение "It is known that it is false that varenyky eats me." равносильно предложению "It is known that varenyky doesn't eat me.".

Предложение "It is known that it is false that I don't eat varenyky." равносильно предложению "It is known that I eat varenyky.".

2. It is known that [the sentence] "Abel and Cain killed each other." implies [the sentence] "Abel killed Cain.". However, it is known that it is false that Abel killed Cain. Therefore, it is known that it is false that Abel and Cain killed each other.

Примечание

Предложение "It is known that "Abel and Cain killed each other." implies "Abel killed Cain."." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложение "It is known that it is false that Abel killed Cain." и предложение "It is known that it is false that Abel and Cain killed each other." - это предложения Английского предметного языка с верификаторами "It is known that...".

Предложение "It is known that it is false that Abel killed Cain." равносильно предложению "It is known that varenyky Abel didn't kill Cain.".

Предложение "It is known that it is false that Abel and Cain killed each other." равносильно предложению "It is known that Abel didn't kill Cain, or Cain didn't kill Abel.".

3. It is known that [the chemical sentence] "C _(s) + O_{2 (g)} → CO _(g)" implies [the chemical sentence] "C + O₂ = CO". However, it is known that it is false that C + O₂ = CO. Therefore, it is known that it is false that C _(s) + O_{2 (g)} → CO _(g).

Примечание

Предложение "It is known that "C _(s) + O_{2 (g)} → CO _(g)". implies "C + O₂ = CO".." - это предложение Английского метаязыка с верификатором "It is known that ...".

Предложения "It is known that it is false that C + O₂ = CO." и "It is known that it is false that C _(s) + O_{2 (g)} → CO _(g)." - это предложения Английского предметного языка с верификаторами "It is known that...".

Формулировка

It is known that if $~{\mathfrak {a}},$ [then] ${\mathfrak {b}}.$ [However,] it is known that it is false that $~{\mathfrak {b}}.$ $~\Rrightarrow$ It is known that it is false that $~{\mathfrak {a}}.$

Examples

1. It is known that if I don't eat varenyky, [then] varenyky eats me. However, it is known that it is false that varenyky eats me. $~\Rrightarrow$ It is known that it is false that I don't eat varenyky.

2. It is known that if Abel and Cain killed each other, [then] Abel killed Cain. However, it is known that it is false that Abel killed Cain. Therefore, it is known that it is false that Abel and Cain killed each other.

3. It is known that if C _(s) + O_{2 (g)} → CO _(g), [then] C + O₂ = CO. However, it is known that it is false that C + O₂ = CO. Therefore, it is known that it is false that C _(s) + O_{2 (g)} → CO _(g).

Формулировка

It is known that $~{\mathfrak {b}}$ if $~{\mathfrak {a}}.$ [However,] it is known that it is false that $~{\mathfrak {b}}.$ $~\Rrightarrow$ It is known that it is false that $~{\mathfrak {a}}.$

Examples

1. It is known that varenyky eats me if I don't eat varenyky. However, it is known that it is false that varenyky eats me. $~\Rrightarrow$ It is known that it is false that I don't eat varenyky.

2. It is known that Abel killed Cain if Abel and Cain killed each other. However, it is known that it is false that Abel killed Cain. Therefore, it is known that it is false that Abel and Cain killed each other.

3. It is known that C + O₂ = CO if C _(s) + O_{2 (g)} → CO _(g). However, it is known that it is false that C + O₂ = CO. Therefore, it is known that it is false that C _(s) + O_{2 (g)} → CO _(g).

См. также

Користувач:Галактион/Modus ponens

Галактион 07:03, 19 серпня 2009 (UTC)